There is a good strategy for this in Faster Auto Scaling in AWS CloudFormation Stacks with Lambda-backed Custom Resources
To orchestrate this process, you bootstrap a reference instance with a
user data script, use wait conditions to trigger an AMI capture, and
finally create an Auto Scaling group launch configuration that
references the newly created AMI. The reference instance that is used
to capture the AMI can then be terminated, or it can be repurposed for
administrative access or for performing scheduled tasks.
The process does not use Packer and does not require a dedicated server for creating the AMI, and instead uses a Lambda-backed custom resource.
Second Option
As you already have your AMI creation in Packer working, you should consider using Lambda to copy your existing Launch Configuration with the updated AMI. You can see a good approach to this from Patch an AMI and Update an Auto Scaling Group:
The following example builds on the Simplify AMI Patching Using
Automation, Lambda, and Parameter Store example by adding a step that
updates an Auto Scaling group with the newly-patched AMI. This
approach ensures that new images are automatically made available to
different computing environments that use Auto Scaling groups.
The final step of the Automation workflow in this example uses an AWS
Lambda function to copy an existing launch configuration and set the
AMI ID to the newly-patched AMI. The Auto Scaling group is then
updated with the new launch configuration. In this type of Auto
Scaling scenario, users could terminate existing instances in the Auto
Scaling group to force a new instance to launch that uses the new
image. Or, users could wait and allow scale-in or scale-out events to
naturally launch newer instances.